Effective-mass theory ofp-type heterostructures under transverse magnetic fields

Abstract

We present, within the transfer-matrix formalism, a second-order k⋅p effective-mass theory for p-type heterostructures under transverse magnetic fields (parallel to the layers). The Luttinger Hamiltonian of ${\mathrm{\Gamma }}_8$ states is employed to describe valence bands, with the coupling between heavy and light holes included. We expand the envelope functions in parabolic cylinder functions, and reduce the effective-mass equations from coupled differential equations to matrix equations. The matrices involved are of small dimensionality (typically 8×8), allowing for treating hole magnetic levels and magnetotunneling in a concise manner. The theory has been applied to a single-barrier structure and a double-barrier structure, both made of GaAs and ${\mathrm{Al}}_{\mathit{x}}$ ${\mathrm{Ga}}_{1\mathrm{-}\mathit{x}}$As. In the former case, we present valence bands and wave functions, and, in the latter, we show hole transmission under transverse magnetic fields for various channels. The model presented here is well suited to magneto-optic- and magnetotransport-property calculations.